Establishment of THz dispersion model of metals based on Kramers-Kronig relation

The extraction of terahertz dispersion parameters is confined in a limited region due to the limitation of the existing THz techniques. A method of studying the dispersion model of metals from the measurements of reflection spectrum and analysis of Kramers-Kronig (KK) relation is proposed. The reflection spectrum is measured by Vertex 80V Fourier transform spectrometer. In order to eliminate the signal noise of measured reflection spectrum, the measured spectrum is smoothed by Drude estimation. Using the smoothed reflection spectra of copper (Cu) alloy and aluminum (Al) alloy in a range of 4-40 THz, the complex refractivities are inversed based on the KK relation of amplitude and phase of reflective coefficient. The constant extrapolations at lower frequencies and the exponential extrapolation at higher frequencies are adopted in the KK integration. The exponential extrapolation index is adjusted according to the calibrating complex refractivity measured from far-infrared ellipsometer. According to the inversed complex refractivity, the plasma frequency and damping frequency in Drude model are optimized using the genetic algorithm. The objective function is defined as the error between the fitted complex refractivity and KK inversion. Since the optimal plasma frequency and damping frequency are different for different fitting frequencies, the obtained Drude parameters are averaged in order to reduce the influences of errors from KK inversion, measured reflection spectrum and calibrations. The complex refractivity indexes in a range from 15 THz to 40 THz, calculated by the established Drude model, are in good agreement with the measured calibrations from ellipsometer, which demonstrates the accuracy of the established Drude dispersion model. The reflection spectra below 4 THz are greatly distorted due to the signal noise, and the calibrating refractivity is located in the far infrared region, thus the complex refractivity is inversed in a region of 4-40 THz by KK algorithm. The complex refractivity indexes in a range of 0.1-20 THz, obtained by the proposed scheme, are for the vacancy, which will provide great support for the dispersion analysis in the whole terahertz gap. The procedures are helpful for extrapolating the dispersion information to terahertz band from the far infrared region. The scheme takes the advantage of the spectrometer and ellipsometer, and it requires high experimental precisions of reflection spectrum and calibrating refractivity. In addition, the scheme is adaptive to both metals and nonmetals by applying proper dispersion model which depends on the property of the reflection spectrum. The established model determines the microscopic dispersion parameters of material, which provides great support for the investigation of terahertz dispersion analysis, scattering mechanisms and imaging processes.